After we have connections on a vector bundle, we may consider its curvature just as what we did in Riemannian geometry. Furthermore, using the curvature, the Chern class is introduces to characterze vector bundles on a manifold. However, the Chern class turns out to be independent of the connection and hence gives us an important tool in the study of the topology of manifolds and bundles.
Curvature of a connection
Recall that a connection on a complex vector bundle is given as a sheaf homomorphism We see that is not a homomorphism of -modules, as is required to satisfy the Leibniz’ rule The connection can be extended into sheaf homomorphisms by where and are given on an open subset . This is well-defined as we can verify that We see that the generalized Leibniz’ rule is satisfied, that is for any and , we have
Thus we may consider the decompostion of with itself For a local section of and a local smooth function , we have This shows that is a homomorphism of -modules. Consequently, may be identified with a global section of the bundle , where is the bundle of endomorphisms of and is usually isdentified with . The -valued -form is called the curvature of the connection .
Suppose form a local frame of and with respect to this frame the connection matrix of is , i.e., We can express as a matrix of -forms locally by Direct computation yields that Thus i.e., The matrix is called the curvature matrix of with respect to the local frame . Consider another local frame with corresponding curvature matrix . If the relations between and is given by the matrix , i.e., then we can compute that This verifies the tensorial property of and .
Additional properties of the vector bundle and the connection yields addition properties of the curvature . If is a holomorphic vector bundle and is a complex connection, then considering a holomorphic frame of yields that , i.e., . If is further a hermitian vector bundle, then we may consider an orthonormal frame . With respect to this frame, we have and hence i.e., Thus implying that . Thus is of type and is skew-hermitian.
Given connections and curvature on some vector bundles, we can consider the induced connections and curvature on induced bundles, including direct sum, tensor product, dual bundle, and pullback.
Suppose and are vector bundles on with corresponding connections and curvature . Then we have a natural connection on the direct sum given as The corresponding curvature is clearly
Similarly we have a natural connection on the tensor product given as where and We can verify that the curvature of can be given as where we use similar identifications as above.
Now suppose is a vector bundle on with connection and curvature . The induced connection on the dual bundle is given by the formula where is a local section of and is a local section of . Thus With respect to a dual frame, we see that .
Further suppose is a smooth map. Then the pullback defines a connection on the pullback bundle . If is the connection matrix of with respect to the frame on an open subset , then the connection matrix of with respect to the frame on the open set is exactly . Thus we can see that the curvature of is nothing else than .
Chern classes of a vector bundle
In order to define the Chern classes of a vector bundle, we need to discuss a bit about functions of matrices which are invariant under conjugation.
Suppose is a homogeneous polynomial function in the entries of degree . We call an invariant polynomial if holds for any and .
The basic examples of such polynomials are the elementary symmetric polynomials of the eigenvalues of . Consider the polynomials given by the formula In particular, we have , , and is the sum of the determinants of -principal minors of . These are called elementary invariant polynomials.
In fact, any invariant polynomial can be expressed as a polynomial of these elementary invariant polynomials. Suppose is an invariant polynomial. Consider the polynomial which is symmetric in . Then there exists a polynomial such that where is the elementary symmetric polynomial in of degree . It follows that holds for any diagonalizable matrix , and hence holds for any matrix .
Now consider an -linear form We say that is invariant if holds for any and . We see that each invariant -linear form gives an invariant polynomial of degree by Conversely, it can be verified that each invariant polynomial of degree admits a symmetric invariant -form such that and are related as above. Such is actually unique and is called the polarization of .
Suppose is a complex vector bundle of rank and is a connection on with curvature . Consider an open subset with a local frame of on . With respect to this frame, the connection has a connection matrix and a curvatire matrix . Noting that the wedge product of forms of even degree is commutative, the expression gives a well-defined -form on for each invariant polynomial of degree . Since the curvature matrix with respect to different frames only differ by a conjugation, we see that with local expression is a well-defined -form on .
Theorem. Suppose is a complex vector bundle on a smooth manifold and is the curvature of a given connection on . If is an invariant polynomial of degree , then and the cohomology class in the de Rham cohomology group is independent of the choice of the connection on .
Before the proof of the theorem, we need some preparations concerning the extension of the connection and some further properties of the invariant -forms.
Recall that the connection induces a natural dual connection by the formula Then we have a connection as we have a natural connection on the tensor bundle . This actually extends to sheaf homomorphisms
Lemma (Bianchi identity). Suppose is a connection on the complex vector bundle with the curvature . Then
Proof: The action of the connection on the endomorphism bundle can be verified to be given by where is a local section of and is a local section of . It follow that for any local section ,
Suppose is a symmetric -linear invariant form, where is the rank of . Using the invariance of , we obtain a well-defined symmetric -linear invariant form This further induces a natural -linear map Moving to the section sheaf, we obtain a sheaf homomorphism
Lemma. Suppose is a connection on the complex vector bundle of rank and is a symmetric -linear invariant form on . Then for any -valued forms , we have
Proof: By the explicit formula of the extension of to , we can see it suffices to show the case , i.e., to show that where This can be treated locally, where we may assume the connection matrix of is given by . Then direct computation yields that where is an matrices of smooth functions. Since the Leibniz’ rule implys that it remains to show By the multi-linearilty, we only need to show that holds for a normal matrices .This formula is just a corollary of the invariance of if we consider the derivative of the constant function
Now we can prove the theorem.
Proof of the theorem: Suppose is the polarizaton of . Then we have by the preceding two lemmas, showing the first assertion.
To prove the second assertion, assume that is another connection on . For each , we can see that also defines a connection on . Suppose the curvature of the connection is . It suffices to show that is always a -exact -form. By direct computation in local frame we can verify that which impies that
The above theorem gives for each complex vector bundle a homomorphism from the graded algebra of invariant polynomials to the graded algebra given by where is the curvature of any connection on . This homomorphism is called the Chern-Weil homomorphism.
Let denote the elementary invariant polynomial of degree . We define the Chern form of the connection on by where is the curvature of . The Chern class is then defined by The total Chern class is the sum of the Chern classes where we set . If is a complex manifold, then we take the Chern classes of to be the Chern classes of its holomorphic tangent bundle .
By the independence of the Chern classes with the connection, we can see the Chern classes must be real. Indeed, consider an hermitian metric on and a connection on compatible with the hermitian structure. Then the corresponding curvature is skew-hermtitian, i.e., . Hence we have implying that .
Suppose and are complex vector bundles on , with curvature and . The curvature on the direct sum is given by and hence Thus
Suppose is a complex vector bundle of rank on with curvature and is a line bundle on with curvature . Then the curvature of is given by It follows that More generally, if is a complex vector bundle of rank , then
The curvature of the dual bundle is given by implying that It follows that
If is a smooth map, then the curvature on the pullback bundle is exactly . Thus the Chern classes of is given as
On vector bundles (not neccesarily holomorphic) on a complex manifold, we may consider metrics and connections analogously to the real case. These induce the notion of curvature and Chern class and turn out to be important tools in complex geometry.
Hermitian metric on vector bundles and hermitian manifolds
Suppose is a complex vector bundle on a smooth manifold . Then a hermitian metric on is a hermitian inner product on each fiber depending smoothly on . More precisely, for any open subset and smooth sections of on , the map is a smooth function on . A complex vector bundle together with a hermitian metric is called a hermitian vector bundle.
Using local frame of on an open subset , a hermitian metric on is equivalent to a positive definite hermitian matrix consisting of smooth functions on . Applying the partition of unity theorem, we see that any complex vector bundle on a smooth manifold admits a hermitian metric.
Now suppose is a complex manifold. If the holomorphic tangent bundle of admits a hermitian metric , then we also call a hermitian metric on . A complex manifold together with a hermitian metric is called a hermitian manifold.
For each point , the hermitian metric gives a hermitian form or equivalently
Thus can be identified with a section of the vector bundle . If is a local coordinate of , then the hermitian metric can be expressed as where
Consider the composition with the natural -linear isomorphism given by for each , we obtain a bilinear form This bilinear form can also be constructed by viewing as a subspace and extending the hermitian form on to by zero. Thus the real part of gives a symmetric bilinear form which is further an inner product by the positive definite property of , while the imaginary part of gives an alternating form which determines an element in . Therefore we obatin a Riemannian metric on , called the associated Riemannian metric of , and a real differential -form on , called the fundamental form of . Noting that can also be viewed as an alternating hermitian form on for each , we see that is also a -form, i.e.,
We consider the explicit expression of and in local coordinates. Direct computation yields that and that
Suppose is a holomorphic map such that the holomorphic tangent map is injective for any . Then we may define a hermitian metric on by Viewing a hermitian metric as a section of the tensor bundle, we see that is exactly the image of under the pullback Suppose is the fundamental form of . Then it can be verified that .
Consider a local coordinate of such that form an orthonormal basis locally. Such local coordinate exists by the Gram-Schimdt process. With respect to this basis, we see that The associated volumn form to the Riemannian metric is given by Direct computation then yields that Now suppose is a -dimensional complex submanifold of . Then we have the following formula, which is called the Wirtinger theorem,
We could say more about the fundamental form of a hermitian metric. Indeed, any real -form on determines a hermitian form on . Suppose we have a real -form on expressed locally as Then form a hermitian matrix and hence determines a hermitian form on locally. As is actually independent of coordinates, this hermitian form should be independent of coordinates as well. If the corresponding matrix is positive definite everywhere, then actually defines a hermitian metric on . Such is called a positive -form, and we see a correspondence between hermitian metrics and positive -form on .
We end this part with some examples.
The hermitian metric on given as is called the Euclidean metric. The associated Riemannian matric is exactly the Euclidean metric on .
Suppose is a lattice generated by real-linearly independent vectors. Then there is a standard hermitian metric on the complex torus given by which is also called the Euclidean metric.
There is a hermitian metric on called the Fubini-Study metric constructed as follows. Note that it suffices to give a positive -form on . Consider the real -form on given by If is a lifting of the covering map on an open set , then is a real -form on . For two lifting and on and , respectively, there is a nonvanishing holomorphic function such that on , implying that on the intersection Thus these pullbacks glue together to a real -form on the whole . To see that is positive, noting that acts transitively on and leaves invariant, it suffices to show gives a postive definite hermitian form at one point. Using the local coordinate around , we see that at , is given as which is clearly postive definite.
Connections on vector bundles
In analogy to real case, we need connections on vector bundles to compare different fibers.
Suppose is a complex vector bundle on a complex manifold . Consider the sheaf of smooth sections of the tensor bundle , whose sections are usually called -valued -forms, and the sheaf of smooth sections of the tensor bundle , whose sections are called -valued -forms. Noting that is exactly the sheaf of smooth sections of .
A connection on is a sheaf homomorphism satisfying the Leibniz’ rule for any and on an open subset .
We can express a connection in a local frame. Suppose form a local frame of , i.e., they are sections of on an open subset such that form a basis of for each . Then there are differential -forms on such that We obtain a matrix of -forms on , called the connection matrix of with respect to the local frame . Conversely, given a matrix of -forms, we may define a connection on by
The connection matrix depends on the choice of the local frame. Suppose is another frame, with corresponding connection matrix . The relations between two frames is given by a transformation matrix such that Then we have implying that This is the transformation formula of connection matrices.
Using the connection, we can consider the directional derivatives of sections. Suppose is a connection on a complex vector bundle on . For , a section of on a neighborhood of , and a tangent vector , the directional derivative of at along the direction of is given by where the pairing comes from the natural pairing If is instead a tangent vector field near , then gives a section of near .
Suppose is a smooth curve from to . For each , we can define the tangent vector of the curve at to be
If is a section of on a neighborhood of this curve, then the value of only depends on the value of on . If is a section of on such that , then we call a section of parallel along , and the vector is called the parallel displacement is along .
The choice of connections on a vector bundle is in general not canonical. However, if we consider more structure on the vector bundle and require the connection satisfying some compatible condition, then the choice becomes unique.
First suppose is a holomorphic vector bundle on a complex manifold . Using the decomposition we have Thus we may write a connection as with
Note that since is a holomorphic vector bundle, gives a well-defined sheaf homomorphism Indeed, suppose is a local holomorphic frame of and is an -valued -form on , then we can write as We define to be the section To check this is independent of the choice of the frame, consider another holomorphic frame with Then the coefficients are holomorphic functions on . With respect to this new frame, we have and hence This shows that is well-defined.
In particular, we have a sheaf homomorphism A connection on is called a complex connection, i.e., a connection compatible with the holomorphic structure, if . Using this definiton it is not hard to verify the following proposition.
Proposition. Suppose is a holomorphic vector bundle on a complex manifold and is a connection on . Then is a complex connection if and only if with respect to any local holomorphic frame of , the connection matrix of consists of -forms.
Now suppose is further a hermitian holomorphic vector bundle on with hermitian metric . Then induces a hermitian inner product at each . If is a connection on such that the parallel displacement induced by always preserves the inner product of two holomorpic tangent vectors, then we say is compatible with the hermitian metric . We may also verify that is compatibe with if and only if for each holomorphic sections of , we have where the pairing
and
comes from the pairing
Theorem. Suppose is a holomorphic vector bundle on a complex manifold with a hermitin metric . Then there is a unique complex connection on compatible with , called the metric connection on .
Proof: Suppose is a local holomorphic frame of , and let . Assume that such connection exists, and the connection matrix of with respect to is . Then must be -forms. Consider the inverse matrix of , that is Since is compatible with , we have the equation or equivalently, This has a unique solution given by Thus we have a uniquely determined connection on satisfying all the condition.
In this post we focus on holomorphic line bundles and discuss some relations between divisors and line bundles.
Definition of divisors
Suppose is an -dimensional complex manifold and is an analytic subvariety of dimension . For each , there is a neighborhood of such that is defined as the zero set of a holomorphic function on this neighborhood. This functions is called a local defining function for near , and is unique up to multiplication by a function nonzero at .
Proposition. Suppose is a complex manifold and is an analytic hypersurface on . Then for any connected component of , the closure is an analytic variety.
Corollary. An analytic hypersurface is irreducible if and only if is connected.
We see from this proposition that each analytic hypersurface can expressed uniquely as the union of some irreducible analytic hypersurfaces.
Now we define a divisor on is a locally finite formal linear combination of irreducible analytic hypersurfaces of . The local finiteness here means that for each there is a neighborhood of intersecting with only finitely many which appear in . The divisors on form an additive group . If holds for all , then we call an effective divisor, written as . Note that if is an analytic hypersurface, then we can identified it with the divisor
Suppose is an irreducible analytic hypersurface, and is a defining function for near . For any meromorphic function defined in a neighborhood of , we can define the order of along at to be the unique integer such that
This is independent of the choice of the defining function and the point .
We say has a zero of order along if , and that has a pole of order along if . We can see that for any two meromorphic functions , it holds that
Now consider a nonzero meromorphic function on . The divisor associated to is given by This divisor can be written as the difference of two effective divisor, namely the zero divisor
and the pole divisor
The divisors can be constructed in sheaf-theoretic terms. We claim that a divisor on is equivalent to a global section of the quotient sheaf of with respect to , and then we have the natural isomorphism
On the one hand, a global section of the quotine sheaf is given by an open cover of and meromorphic functions on that are not identically zero with
Thus for any analytic hypersurface we have
if they are defined. The corresponding divisor is then given by
where is chosen such that .
On the other hand, given a divisor thee is an open cover of such that every has a local defining function in each . We then let
to obtain a global section of .
We see from the above constructions that the identification is actually a homomorphism and hence an isomorphism.
Suppose is a holomorphic map between complex manifolds. Then for each divisor on such that the image of is not contained in the support of , we may define the pullback of along . Suppose under the identification of a divisor with a section, is given by , then is the divisor on given by . For holomorphic functions such that is dense in , we then obtain a homomorphism
Relations between divisors and line bundles
Recall that a line bundle on a complex manifold can be given by an open cover of and transition functions of . Using this description, it is not hard to see that the set of line bundles on can be identified with .
The set of line bundles on can be given the structure of group with multiplication given by tensor product and inverses given by dual bundles. This group is called the Picard group of , denoted by . Note that the group structure of and the group structure of is actually the same, we have the natural isomorphism
Now we attempt to associate a line bundle to each divisor on . Let be a divisor on with local defining functions over an open cover of . Then the transition functions determine a line bundle on , called the associated line bundle of . This is independent of and hence well-defined.
We see the following properties of :
if and are divisors on , then implying that the corresponding map is homomorphism;
the line bundle is trivial if and only if there is a nonzero meromorphic function on such that .
Thus we say that two divisors and are linearly equivalent if for some nonzero meromorphic on , or equivalently , written as .
The above discussion can be interpreted in the sheaf-theoretic opinion. Consider the exact sequence of sheaves on This induces the exact sequence of cohomology groups that Identifying the corresponding cohomology groups with and , respectively, we see that the homomorphism maps each meromorphic function to the divisor , and the homomorphism maps each divisor to the line bundle . Hence the latter property of the associated line bundle is nothing else than the exactness of the sequence.
Holomorphic and meromorphic sections of line bundles
Suppose is a holomorphic line bundle on a complex manifold with trivializations and corresponding transition functions . These trivializations induce ismorphisms and then give a correspondence Thus a section of on is equivalent to a collection of functions satisfying on each
According to this point of view, we define a meromorphic section of on to be given by a collection of meromorphic functions satisfying on each . This is equivalent to the section on of the sheaf Note that the quotient of two nonzero meromorphic sections of is a well-defined meromorphic function on .
For a non-trivial global meromorphic section of , we have implying that for any irreducible hypersurface , we have Thus we can define the order of along by The divisor associated to is then given by It is clear that is holomorphic is and only if is effective.
Proposition. Suppose is a complex manifold. Then the image of the natural map consists of those line bundles admitting non-trivial meromorphic sections.
Proof: If is given by functions , then clearly these functions give a meromorphic section of with . Conversely, suppose is given by transition functions and is a non-trivial global meromorphic section of , then and hence .
The holomorphic sections of the line bundle associated to a divisor on may be constructed in the following way. Consider the set of meromorphic functions on such that Fixed a global meromorphic section of with . On the one hand, any holomorphic section of induces a meromorphic function on such that On the other hand, any meromorphic function gives a holomorphic section of . Thus we obtain a bijection
We end this post with some discussion of the relations between holomorphic sections of line bundles and holomorphic maps into projective spaces.
Suppose is a holomorphic line bundle on a complex manifold and are global holomorphic sections of having no common zero. Suppose is an open cover of with trivializations on each and transition functions . On each , the sections can be expressed by holomorphic functions . Then we have a map given by On , we have suggesting that we may define a well-defined function by for . This map is clearly holomorphic.
Conversely, assume we have a holomorphic map . Then we may take an open cover of such that can be expressed as with having no common zero. By the definition of homogeneous coordinate, there are nonvanishing holomorphic functions such that Using these as transition functions, we obtain a holomorphic line bundle on . Meanwhile, the holomorphic functions define a holomorphic section of for each , and these sections have no common zero.
We conclude that a holomorphic map from to is equivalent to holomorphic sections of a line bundle on with no common zero.
Holomorphic vector bundles are quite essential objects in the study of the geometry of a complex manifold. On one hand, using holomorphic vector bundles, we can consider the analogy of Riemannian geometry in the complex case. On the other hand, the study of vector bundles leads to the concept of divisors, which are frequently considered in algebraic geometry as well.
Definitons and constructions
The definition of a holomorphic vector bundle is analogous to the definition of a smooth vector bundle. Suppose is a complex manifold. A holomorphic vector bundle (of rank ) on is a complex manifold together with a holomorphic map satisfiying that
each fiber where is an -dimenional complex vector space;
for each , there is an open neighborhood of in together with a holomorphic homeomorphism such that is mapped linearly isomorphically onto .
A holomorphic vector bundle of rank is usually called a holomorphic line bundle.
Suppose and are holomorphic vector bundles on . Then a homomorphism from to is a holomorphic map such that for each and that is linear with rank independent of . If is linear isomorphism for each , then we say is an isomorphism, and and are isomorphic.
Note that if and are open subsets in with noempty intersection and trivializations then we have a holomorphic map
given by
These are called transitions functions of . The transition functions of necessarily satisfy the identities
Conversely, given an open cover of and holomorphic maps
satisfying the above identities, we may consider the complex manifold which has the structure of a holomorphic vector bundle on such that are the transition functions. This holomorphic vector bundle is unique up to an isomorphism.
Suppose and are holomorphic vector bundles of rank and on with transition functions given by and . Using the description of a holomorphic vector bundle by transition functions, we have the following constructions.
The direct sum is given by transition functions
The fiber is canonically isomorphic to .
The tensor product is given by transition functions
The fiber is canonically isomorphic to .
The dual bundle is given by transition functions
The fiber is canonically isomorphic to the dual space of .
The exterior product is given by transition functions
We also have the canonical isomorphism of fibers .
In particular, is the line bundle given by transition functions called the determinant line bundle of .
Suppose for each the matrix can be written as
then is naturally a holomorphic subbundle of . The quotient bundle is then given by the transition functions .
Suppose is a vector bundle homomorpism, then there exist holomorphic subbundles and such that we have the canonical isomorphism of fibers These are called the kernel bundle and image bundle of . We can also define the cokernel bundle of , which is exactly the quotient bundle Using these definitions, we may consider the exact sequences of vector bundles in a natural way.
Suppose is a holomorphic map between complex manifolds and is a holomorphic vector bundle on with transition function . The pullback bundle of along is the holomorphic vector bundle on given by transition functions . For each there is a canonical isomorphism
If is a submanifold of and is the inclusion, then we call the restriction of on .
For a holomorphic vector bundle , we define a holomorphic section of on an open subset to be a holomorphic map such that is identity on , i.e., for each . The collection of all holomorphic sections of on is denoted by . The assignment gives a sheaf on , denoted by . We see that is naturally a -module.
Tangent bundles and related bundles
Suppose is an -dimensional complex manifold. From its complex manifold structure, we may construct some holomorphic vector bundles on which are not trivial in general.
Viewing as an -dimensional smooth manifold, we can consider its tangent bundle and its complexification . Note that the fiber of at is exactly
Then we have smooth subbundles of given by
and
The bundle is called the holomorphic tangent bundle of and is called the antiholomorphic tangent bundle of .
We claim that the holomorphic tangent bundle is actually a holomorphic vector bundle on . Consider a coordinate covering of . The transition functions of with respect to the trivializations induced by the coordinate maps are given by These are holomorphic, and hence is a holomorphic vector bundle. We may also write for the holomorphic tangent bundle of to specify that it is a holomorphic vector bundle.
Similarly we can define the holomorphic cotangent bundle and the antiholomorphic cotangent bundle on , and is a holomorphic vector bundle, also denoted by . Note that is the dual bundle of .
Using , we can defined the bundle of holomorphic -forms on . In particular, is called the canonical bundle of . We can see that the transition functions of is given by
Analogously to the construction of -forms, we may also construct the bundle of -forms on given as The smooth sections of are exactly the -forms on .
Now suppose is a complex submanifold of . Using the local expression of a complex submanifold, we see that the holomorphic tangent bundle of is naturally a subbundle of . Then we define the normal bundle of in by the following exact sequence which called the normal bundle sequence:
Proposition (Adjunction formula). Suppose is a complex submanifold of . Then the canonical bundle of is naturally isomorphic to the line bundle .
This is actually a corollary of the following result, which can be shown by considering the relations of the transition functions.
Proposition. Suppose the following sequence of vector bundles is exact: then there is a canonical isomorphism
In analogy to the de Rham cohomology, we define the Dolbeault cohomology for complex manifolds. To compare this new cohomology with the sigular cohomology, the sheaf theory is applied. It is worth reminding that the sheaf theory is of great importance in the further study of complex geometry.
Dolbeault cohomology
Suppose is a complex manifold. Let be the kernel of the linear map The forms in are said to be -closed. Since , we have The Dolbeault cohomology group (of order ) of is then defined to be the quotient space
We may consider the functorial property of the Dolbeault cohomology. Suppose is a holomorphic map between compex manifolds. The pullback of differential forms gives maps As is holomorphic, it can be verified that
Noting that commutes with , we obtain the induced homomorphisms
Lemma. Suppose is an open neighborhood of the closure of a bounded disc . Consider . Then the function is well-defined on and satisfies .
Proof: Consider any . By the existence of the partition of unity, we may take the decomposition with supported on and supported outside . Let It is clear that is well-defined for . Consider the change of variable , we may write which is certainly well-defined. Thus is well-defined for .
Since is holomorphic in for in the support of , we see for that Meanwhile, using the above expression of , we can compute that For a fixed and any sufficiently small , we obtain from the Stokes’ formula that Letting , this yields Since is arbitrary, we see that on the whole .
Lemma. Suppose is an open neighborhood of the closure of a bounded polydisc and . If is -closed, then there exists such that on .
Proof: It suffices to prove for . Suppose is given as We claim that if the decomposition of does not involve any for , then there exists such that does not involve any for . This is sufficient since the conclusion can be shown by induction after this assertion.
Let Then Comparing the terms of and , we have for any and such that .
Consider the functions Let Then the preceding lemma implies that this is the desired up to a sign.
We procedure by induction on . Suppose has been constructed. Take any such that holds on . Then holds on . When , there is some such that on . The can then be given by
Now consider the case . We see that is holomorphic on . Consider the power series expansion of this function and truncate it to get a polynomial such that Then we can set
In both cases we obtain a series such that exists and satisfies on .
Sheaf theory on complex manifolds
The definitons and basic results in sheaf theory can be found in this PDF file.
There are many examples of sheaves on a complex manifold :
the locally constant sheaves ;
the additive sheaf of holomorphic functions;
the multiplicative sheaf of nonvanishing holomorphic functions;
the sheaf of holomorphic -forms, which can be expressed as with holomorphic;
the sheaf of -forms;
the sheaf of -closed -forms;
the sheaf of holomorphic functions vanishing on a fixed analytic subvariety .
Suppose is an open subset of . A meromorphic function on is given locally as the quotient of two holomorphic functions. Precisely, there is a covering of such that the restriction of on is given by for each , where and are relatively prime in and in . We can then consider the sheaf of meromorphic functions and the multiplicative sheaf of nonzero meromorphic functions.
The exact sequences of sheaves on are widely used in the study of complex manifolds.
The exponential morphism yields an exact sequence
This is called the exponential sheaf sequence.
Suppose is a submanifold. The sheaf can be viewed as a sheaf on . Then the sequence is exact, where the morphisms are given by inclusion and restriction.
For a sheaf on a topological space we can consider its cohomology groups . Basic results concerning the sheaf cohomology can also be found in this PDF file.
Applying the results of sheaf theory to a complex manifold , we obtain the following theorem.
Theorem (Dolbeault theorem). Suppose is a complex manifold. Then we have the canonical isomorphism for each nonnegative and .
Proof: Note that the exact sequence gives an acyclic resolution of the sheaf , we have the canonical isomorphism
Another application of the sheaf theory is the answer to the Cousin problem.
Proposition. Any analytic hypersurface in is the zero set of an entire function.
Proof: Suppose is an analytic hypersurface. For each , there is a neighborhood of such that is given by a holomorphic in this neighborhood, and can be chosen to be of no square factor uniquely up to a unit.
Thus there is a cover of and functions such that for each , and that
for each and . Since
after a refinement of covering if necessary, there exists for each such that . Therefore we obtain an entire function whose zero set is exactly .
In this post we first study some properties of a holomorphic map from its Jacobian, and then introduce the concept of submanifolds and subvarieties. After that we disuss a bit about differential forms, which is a preparation for calculus on a complex manifold.
Inverse function thorem and implicit function theorem
In analogy to the real case, we have the following two standard results.
Theorem (Inverse function theorem). Suppose is an open subset and is holomorphic. If is nonsingular at , then there exists a neighborhood containing and a neighborhood containing such that is a holomorphic homeomorphism.
Proof: Since at , the real inverse function theorem yields a smooth inverse of near . It remains to show the holomorphic property of . As , we have for any . It follows from the nonsingularity of that for each and , implying that is holomorphic.
Theorem (Implicit function theorem). Suppose is an open subset, and is holomorphic. If satisfies that then there exist open subset and a holomorphic map such that and
Proof: Again the real implicit function theorem yields a smooth function satisfying the required property. To show the holomorphic property, note that for , which implies for any and .
However, we also have some special features of the complex case.
Theorem. Suppose is a bijective holomorphic map between two open subsets . Then is nonvanishing. In particular, is a holomorphic homeomorphism.
Proof: Prove by induction on . The case when is proved in the reviewing post of holomorphic functions of one variable. Suppose the assertion is proved for any . Consider any such that . We claim that . Assume that . Then we may suppose that is nonsingular. By the inverse function theorem, form a local coordinate system around . It is clear that maps bijectively to . However, the Jacobian of the restriction of to is singular at , which contradicts the induction hypothesis. We conclude that we must have .
By the above discussion, we see that is constant on each connected component of . Since Weierstrass preparation theorem tells us that has positive dimension locally if it is nonempty, the injectivity of implies that is nonvanishing on .
Submanifolds and subvarieties
Like the real case, we can consider submanifolds of a compplex manifold. Suppose is a complex manifold of dimenional . Then a -dimensional complex submanifold is a subset of satisfying that there is a collection of holomorphic coordinate charts of covering such that for each ,
By the inverse function theorem, we can see that this is equivalent to that is given by the zero sets of holomorphic functions such that .
The idea of express a subset as the zero set of some holomorphic functions gives us the concept of subvarieties. An analytic subvariety of a complex manifold is a subset given locally as the zero set of a finite collection of holomorphic functions. A point is called a smooth point or a regular point if is given in a neighborhood of by holomorphic functions with . Denote the set of regular points on by , and let . The points in are called sigular points of . An analytic variety is irreducible if it cannot be written as the union of two proper analytic subvarieties.
We can see that each connected component of is a complex submanifold . There is a theorem saying that an analytic variety is irreducible if and only if is connected. Thus we can define the dimension of an irreducible variety to be the dimensional of .
Differential forms on a complex manifold
The last part of this post devotes to some discussion of -forms on a complex manifold . Viewing as a smooth manifold, we can consider the space of -forms on . Let Then the exterior differentiation gives a linear map
For each point , the decomposition
induces a decomposition
Correspondigly, we obtain the decomposition where consists of -forms satisfying for each . A form is said to be of type , and is also called a -form on .
Let be the projection of onto . Define Using local coordinates, we may consider a -form and direct computation yields that and
We can verify that the operators and have the following properties:
;
and ;
for and , we have
We see that and have the similar properties to the exterior differentiation . In next post, they will be used to build a holomorphic analogy to the de Rham cohomology theory.
Complex manifolds are basic objects in complex geometry, just like smooth manifolds are basic objects in differential geometry. We introduce complex manifolds as an analogy of smooth manifolds. Yet it is worth noting some differences between the complex case and the real case.
Definition and examples
A complex manifold (of dimension ) is a -dimensional smooth manifold with an open cover and coordinate maps , such that is holomorphic on .
Like the real case, we can define holomorphic functions on a complex manifold and holomorphic maps between complex manifolds. A holomorphic map which has a holomorphic inverse is called a holomorphic homeomorphism.
The holomorphic condition is actually a quite strong restriction to functions. This can be seen from the following proposition.
Proposition. Any holomorphic function on a compact connected complex manifold is constant.
Proof: Suppose is holomorphic. Since is compact, the continuous function attains its maximum at some point . Consider a coordinate chart with . Then the maximal module principal on implies that is constant. As is connected, must be constant on the whole .
The above proposition suggests that any compact connected complex manifold cannot be holomorphicly embedded in any . However, we still want to put a compact complex manifold into some good space. That is why we introduce the projective spaces.
Use to specify a point in . Define an equivalence relation on by that if and only if for some . The collection of equivalence classes is called the (-dimensional) projective space, denoted by . The equivalence class containing is denoted by , with called the homogeneous coordinates of .
Similar to the real case, we can show that is an -dimensional complex manifold. Consider the unit sphere . The restriction of the natural projection to gives a continuous surjection onto . Since is compact, is also compact.
The projective space can be viewed as a compactification of . Consider the inclusion given by This inclusion is a holomorphic homeomorphism from to an open subset of , and the complement of its image is naturally identified with . When , we see that is exactly the Riemann sphere.
Another example of a compact complex manifold is a complex torus. Suppose are linearly independent as vectors in a real vector space. Let be the lattice generated by . Then is a compact smooth manifold together with a complex manifold structure induced from the covering map . This is called the (-dimensional) complex torus. We see that when , is diffeomorphic to an ordinary torus .
We try to generalize the construction of the complex torus. Suppose is a topological covering map and is a complex manifold. A desk transformation is a homeomorphism from to itself such that . If each desk transformation on is holomorphic, then we can equip with a well-defined complex manifold structure induced from that of . This is similar to the process that defines a quotient smooth manifold.
Consider the group of self-homeomorphisms on generated by the map . Let be the quotient smooth manifold of this group action. Then the natural projection gives a complex manifold structure on , making into a -dimensional complex manifold, called the Hopf surface. The Hopf surface is a compact complex manifold that cannot be embedded in any projective space .
Tangent spaces and cotangent spaces
Next we introduce the tangent space of a complex manifold. Suppose is a -dimensional complex manifold and . Since is a -dimensional smooth manifold, we can consider the real tangent space of at . Let be the complexified tangent space. An element in , called a complex tangent vector of at , is identified with a -linear derivation on the ring of germs of complex-valued smooth functions on a neighborhood of .
Let be the ring of germs of holomorphic functions around , and be that of antiholomorphic functions, i.e., the functions whose conjugates are holomorphic.
If a complex tangent vector vanishes on , then we call a holomorphic tangent vector, and if vanishes on , then we call an antiholomorphic tangent vector. Then collection of all holomorphic (resp. antiholomorphic) tangent vectors is called the holomorphic tangent space (resp.antiholomorphic tangent space) of at , denoted by (resp.).
We try to show that and are both -dimensional complex linear spaces with
Suppose is a holomorphic coordinate chart of around , with . Then form a (-linear) basis of . Let Then Cauchy-Riemann equations imply that form a basis of , while form a basis of . The direct sum decomposition follows.
We can also construct the corresponding cotangent spaces of at . Let be the dual space of , whose elements are called complex cotangent vectors of at . A complex cotangent vector vanishing on (resp.) is called a holomorphic cotangent vector (resp.antiholomorphic cotangent vector), and then we can similarly define the holomorphic cotangent space and the antiholomorphic cotangent space. Note that can be naturally identified with the dual of , while is identified with the dual of .
If we use the local coordinates around , then form a basis of . The corresponding bases of and are then given by and which are exactly the dual basis of and , respectively.
Tangent maps and Jacobians
Since we have tangent spaces of a complex manifold, we can consider the corresponding tangent maps. Suppose and are complex manifolds of dimensional and , respectively. Then any smooth map with induces a tangent map and then a map However, in general, this does not induce a holomorphic tangent map from to , as well as a antiholomorphic tangent map from to .
Proposition. Suppose and are complex manifolds of dimensional and , respectively, and is a smooth map. Consider the tangent map . Then is holomorphic if and only if .
Proof: Note that is equivalent to that is antiholomorphic whenever is antiholomorphic. The latter is directly equivalent to that is holomorphic.
Now suppose is a holomorphic map. Consider the local coordinates around and the local coordinates around . The holomophic tangent map can be expressed by the holomorphic Jacobian Suppose for and for . With respect to the bases and , the real tangent map is given by the real Jacobian If we pass through the tensor product with to consider the complex tangent map, and change our bases into and , we obtain the complex Jacobian
Since and only differ by a change of basis, we have Moreover, when , we have As an corollary, any complex manifold is orientable, as we can fix a natural orientation on by the -form and pull this back through the holomorphic coordinate maps. The pullbacks agree with each other on the intersections of coordinate neighborhoods and hence give a well-defined orientation on the -dimensional complex manifold .
In this post, we explore some properties of the ring and analytic germs in .
First we can show that the local ring is a noetherian UFD. Note that we have the results in algebra that
if is a UFD, then is a UFD;
if is a UFD and are relatively prime, then there exist and nonzero such that .
Theorem. The local ring is a UFD.
Proof. Prove by induction on . When , is a field and clearly a UFD. Suppose is a UFD. Then is a UFD. By Weierstrass preparation theorem, it remains to show that any irreducible factor of a Weierstrass polynomial in is also irreducible in . However, this follows from the facts that any Weierstrass polynomial can be written as a product of irreducible Weierstrass polynomials, and that any irreducible Weierstrass polynomial is irreducible in . These facts can be proved by direct applications of Weierstrass preparation theorem.
Recall that a commutative ring is called noetherian if every ideal in is finitely generated. Hilbert’s basis theorem tells us that if is noetherian then is also noetherian.
Theorem. The local UFD is noetherian.
Proof. Prove by induction on . The result is trivial when . Now suppose that is noetherian. Then is noetherian. Consider any non-trivial ideal and take a nonzero . Up to a change of coordinates, we may assume with a Weierstrass polynomial and a unit. Then . Since is noetherian, the ideal is generated by finitely many elements . For any , Weierstrass division theorem gives with , and then . It follows that . We conclude that is finite generated.
We can consider the zero set of a germ . In fact, what we consider is a germ of sets. Define an equivalence relation on the collection of subsets of by that if and only if there exists an open neighborhood of such that . An equivalence class of with respect to this equivalence relation is called a germ of sets in . Now we can define to be the germ given by the zero set of . Moreover, we can define for a subset its zero set to be the germ given by the common zero set of functions in . Note that the inclusion, union and intersection of germs are all well-defined. We can see that
Proposition. Suppose is irreducible. If vanishes on , i.e., , then divides .
Proof: We may assume is a Weierstrass polynomial with degree . Since is irreducible, and are relatively prime in , and hence there exists a nonzero such that It follows that if has a multiple root in , then is a zero of . Meanwhile, from the identity theorem we see that has an empty interior.
Weierstrass division theorem yields with a Weierstrass polynomial of degree less than . We must have as . Therefore for any outside , has at least distinct roots, implying that for the degree reason. It follows that is identically zero and divides .
Proposition. If is a holomorphic function on a neighborhood of such that is irreducible in , then is irreducible in for in a sufficiently small neighborhood of .
Proof: We may assume is a Weierstrass polynomial. If is reducible in , then with , and hence This shows that the germ of the set of points such that is reducible in is contained in . It suffices to show that is a proper subset of . This is direct from the preceding proposition.
Proposition. If and are holomorphic functions in a neighborhood of such that they are relatively prime in , then and are relatively prime in for in a sufficiently small neighborhood of .
Proof: Still we may assume are both Weierstrass polynomials. Then they are relatively prime as elements in and hence there exist and nonzero such that . This holds on a sufficiently small neighborhood around .
If have a non-unit common factor in , then divides and hence belongs to . It follows that is zero, and hence and are both zero, a contradiction.
Now we consider the relations between ideals of and germs in . A germ in is called analytic if there exist such that as germs. For a germ in , we let to be the set of elements satisfying .
Using the definitions and the fact that is noetherian, we can see that is an ideal of for any germ , and is an analytic germ in for any , where is the ideal generated by . Moreover, the assignments and are both inclusion-reserving. We also have for any analytic germ in and for any ideal .
An analytic germ is called irreducible if any decompostion such that and are analytic germs implies either or .
Proposition. An analytic germ in is irreducible if and only if is a prime ideal in .
Proof: First suppose that is irreducible and . Then we have and hence . Since is irreducible, we must have either or , which implies either or .
Now suppose that is a prime ideal and with analytic germs. If , then , which suggests that is a proper subset of . If we further have , then we can take and . Since , , contradicting to that is a prime ideal.
The final conclusion of the relations between ideals and germs is given by the analytic version of Nullstellensatz.
Theorem (Nullstellensatz). If is an ideal, then where is the radical of , i.e., the ideal of all elements such that for some .
The complete proof is too involved to be presented, so we will just prove the pricipal ideal case of Nullstellensatz.
Proof (of principal case): Suppose is nonzero and . We may assume with a unit and irreducible. Then it suffices to show that each divides . However, this is clear since vanishes on and hence .
For a holomorphic function of one variable, we can consider the decomposition where is an -order zero of and is holomorphic. We want to obtain a similar description of holomorphic functions of several variables near a zero.
To find such decomposition, we need to figure out a special kind of holomorphic functions which share the same zeros with the original holomorphic functions. Consider the Weierstrass polynomials (in , of degree ), which have the form where are holomorphic functions in on a neighborhood of such that
We see that for any fixed such that the above Weierstrass polynomial is defined, has exactly zeros (with multiplicities) as a holomorphic function in .
Lemma. Suppose is a holomorphic function on a neighborhood of , and is a -order zero of the function in . Then there exists a polydisc and such that for each , has exactly zeros (with multiplicities) on as a holomorphic function in .
Proof: There exists such that is the only zero of on . Then there exists such that for any and .
Let be the number of zeros (with multipicities) of on . The argument principle implies that and hence is holomorphic in on by a lemma from the previous post. Thus must be constant, i.e.,
Theorem (Weierstrass preparation theorem). Suppose is a holomorphic function on a neighborhood of , such that and is a nonzero holomorphic function in . Then there exists a polydisc around , a unique Weierstrass polynomial in and a holomorphic function on such that
Proof: Take the same polydisc and the same as the proof of the lemma. Suppose is a -order zero of . By the lemma, we can let be the zeros of on for each .
By residue theorem we have and hence the left hand side is holomorphic in . It follows that the function defined by is a Weierstrass polynomial in a neighborhood of , such that and have exactly the same zeros on this neighborhood.
For each , there is a unique such that is holomorphic in on and that . We can see from the Cauchy integral formula in and a lemma from the previous post that is also holomorphic in , and hence holomorphic in .
The uniqueness of is clear from the fact that and must have the same zeros. The uniqueness of then follows from the identity theorem.
For each holomorphic function on an open subset , we denote the zero set of by . The Weierstrass preparation actually gives a description of in a neighborhood of a zero. The construction of the holomorphic in the theorem can be modified to a proof of the following Riemann extension theorem.
Theorem (Riemann extension theorem). Suppose is a nonzero holomorphic function on a connected open set and is holomorphic and bounded. Then can be extended to a holomorphic function .
Another Weierstrass theorem is concerned with the Euclid algorithm of holomorphic functions.
Theorem (Weierstrass division theorem). Suppose is a holomorphic function on a neighborhood of and is a Weierstrass polynomial in of degree defined around . Then there exists a sufficiently small neighborhood of , a uniquely determined holomorphic function on and a uniquely determined Weierstrass polynomial in of degree less than , such that
Proof: The unique statement can be proved in a similar way to the unique statement of an ordinary Euclid algorithm. Now we show the existence. Taking a sufficient small neighborhood we may define such that is holomorphic for and . It remains to show that is a Weierstrass polynomial with degree less than the degree of .
Suppose Then we can write where are determined by and holomorphic in and . It follows that where are holomorphic in .
For the preceding theorems, we often need to take smaller and smaller neighborhoods. In convenience, we introduce the concept of germs of functions.
Suppose is a fixed point. Let be the set of holomorphic functions defined in some neighborhood of . Define an equivalence relation on by that if and only if and agree with each other on a sufficiently small neighborhood of . An equivalence class of with respect to this equivalence relation is called a germ of holomorphic functions near , and the collection of all such germs is denoted by .
We can see that has a natural ring structure with identity given by the constant function . It can be verified that the units in are the germs given by the functions such that . This imples that is a local ring, whose unique maximal ideal consisting of the germs given by the functions vanishing at . We may use the same notation for a function and the germ it determines for short.
Now we can formulate Weierstrass theorems using the language of germs.
Theorem (Weierstrass preparation theorem). If satisfies that and that is nontrivial, then can be uniquely decomposed as , where is a Weierstrass polynomial and is a unit in .
Theorem (Weierstrass division theorem). Let and let be a Weierstrass polynomial of degree . Then there exist uniquely determined and a Weierstrass polynomial such that .
The theory of holomorphic functions of several variables is the foundation of complex geometry, which uses as its local model. We try to generalize some results from the one variable case, and explore some different phenomena in the several variables case.
Suppose is a point in and satisfies for each . Then we define the polydisc to be the set consisting of the points such that
For a continuous function defined on a connected open subset , we say is holomorphic on if holds on . Meanwhile, we can define for each , called the partial derivative of with respect to .
Applying the result of the one variable case, we obtain the following formula.
Theorem (Cauchy integral formula). Suppoose is a continuous function such that is holomorphic with respect to each single component at each point in . Then for each the following holds
The above formula can be used to obtain the the power series expansion of a holomorphic function. Suppose is a connected open subset and is holomorphic. Then for each , there is a polydisc such that where
The preceding results enable us to generalize the identity theorem, the maximal module principle and the Schwarz lemma to the several variables case.
Theorem (Identity theorem). Suppose is a connected open subset in and is holomorphic. If vanishes on a neighborhood of some , then equals zero on the entire .
Proof: Let be the subset of consisting of the points on a neighborhood of which vanishes. Then is nonempty and open. By the local power series expansion of holomorphic functions, we see that if and only if all derivatives of vanish at , implying that is also closed in . It follows that is exactly the entire .
Theorem (Maximal module principle). Suppose is a connected open subset in and is holomorphic. If attains a maximal value at , then is constant on .
Theorem (Schwarz lemma). Suppose and let . Suppose is holomorphic with , and for each . Then for each , we have where
Proof: For a fixed , let Applying the one-variable Schwarz lemma to the function given by we obtain that
An interesting phenomenon which is not expected for the one variable case is the following theorem.
Theorem (Hartogs’ theorem). Suppose and and satisfy that for each . Then any holomorphic function can be uniquely extended to a holomorphic map .
We refer to the proof of this theorem in Huybrechts’ book.
Lemma. Suppose is an open subset of , is a neighborhood of for some , and is a holomorphic function. Then the function given by is holomorphic.
Proof: Since is compact, we can cover it with finitely many neighborhoods with such that on each neighborhood the power series expansion of converges uniformly, and hence commutes with the integral. This yields a power series expansion of locally.
Now we turn to the proof of Hartogs’ theorem.
Proof of Hartogs’ theorem: For any such that for each , gives a holomorphic function on the annulus .
Consider the Laurent expansion
Then since the lemma implies that is holomorphic for .
Meanwhile, the function is holomorphic for any when , which suggests that when and . It follows from the identity theorem that holds whenever . (This is where we need )
Define This power series converges uniformly on the disc , as the maximal module of each can only be attained on the boundary, and that the power series converges on the annulus.